Bloch wavenumber identification of periodic structures using Prony’s method

Abstract

Phononic crystals and metamaterials are spatially periodic waveguides that may exhibit wave attenuation bands and, therefore, can be applied as vibration attenuation devices. In this type of structure, the wave speed varies throughout the structural domain and, therefore, the local wavenumber also varies. Methods that allow the identification of representative wavenumbers are of key importance for the experimental characterization of these structures and their attenuation bands. In this context, Prony’s method can be a viable and efficient choice. In the present study, one- and two-dimensional Prony methods are, for the first time, addressed for estimating the Bloch wavenumbers of periodic structures. Bloch wavenumbers may be computed from a single unit cell by applying a periodic boundary condition. We show that the measurements used in Prony’s method must be taken at locations that are equally spaced by multiples of the spatial period of the structure. Otherwise, the spatially varying local wavenumber will cause the method to fail. The Total Least Squares Prony method is used to increase the number of observations per unit cell, as a strategy for noise reduction. This is a novel adaptation of Prony’s method for Bloch wavenumber estimation of periodic structures. A Bayesian approach is also proposed for highly noisy measurements. In the current study, we validate the numerical experiments using the theoretical wavenumber values of a periodic Euler–Bernoulli beam, a Levy-type plate periodic along one direction, and a thin square plate (Kirchhoff’s theory) with free boundary conditions periodic along two directions. The solutions for all proposed structures are obtained using the spectral element method and, for the square plate, with the finite element method. One experiment is performed for a periodic beam. Our simulation results show that the adapted Prony method, combined with the methodology to reduce the noise effect, performs very well for the considered phononic crystals.